NeumannValue

NeumannValue is used within partial differential equations to specify boundary values in functions such as NDSolve.

In NDSolve[eqns,{u1,u2,…},{x1,x2,…}∈Ω], are the independent variables, are the dependent variables, and Ω is the region with boundary ∂Ω.

Locations where Neumann values might be specified are shown in green. They appear on the boundary ∂Ω of the region Ω and specify a flux across those edges in the direction of the outward normal.

is used to specify the flux over the part of the boundary ∂Ω where pred is true, such that holds. is the outward-facing unit normal of ∂Ω. The coefficients g and q can depend on any of the independent variables .

When no boundary condition is specified on a part of the boundary ∂Ω, then the flux term over that part is taken to be , so not specifying a boundary condition at all is equivalent to specifying a Neumann 0 condition.

Any logical combination of equalities and inequalities in the independent variables may be used for the predicate pred.

NeumannValue can be used to specify both Neumann and Robin boundary conditions: